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Lie algebra cohomology

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces[1] by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by Claude Chevalley and Samuel Eilenberg (1948) to coefficients in an arbitrary Lie module.[2]

MotivationEdit

If   is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on  . Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential.

The construction of this differential on an exterior algebra makes sense for any Lie algebra, so it is used to define Lie algebra cohomology for all Lie algebras. More generally one uses a similar construction to define Lie algebra cohomology with coefficients in a module.

If   is a simply connected noncompact Lie group, the Lie algebra cohomology of the associated Lie algebra   does not necessarily reproduce the de Rham cohomology of  . The reason for this is that the passage from the complex of all differential forms to the complex of left-invariant differential forms uses an averaging process that only makes sense for compact groups.

DefinitionEdit

Let   be a Lie algebra over a commutative ring R with universal enveloping algebra  , and let M be a representation of   (equivalently, a  -module). Considering R as a trivial representation of  , one defines the cohomology groups

 

(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor

 

Analogously, one can define Lie algebra homology as

 

(see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right exact coinvariants functor

 

Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem.

Chevalley–Eilenberg complexEdit

Let   be a Lie algebra over a field  , with a left action on the  -module  . The elements of the Chevalley–Eilenberg complex

 

are called cochains from   to  . A homogeneous  -cochain from   to   is thus an alternating  -multilinear function  . The Chevalley–Eilenberg complex is canonically isomorphic to the tensor product  , where  denotes the dual vector space of  .

The Lie bracket   on   induces a transpose application   by duality. The latter is sufficient to define a derivation   of the complex of cochains from   to   by extending  according to the graded Leibniz rule. It follows from the Jacobi identity that   satisfies   and is in fact a differential. In this setting,   is viewed as a trivial  -module while   may be thought of as constants.

In general, let   denote the left action of   on   and regard it as an application  . The Chevalley–Eilenberg differential   is then the unique derivation extending   and   according to the graded Leibniz rule, the nilpotency condition   following from the Lie algebra homomorphism from   to   and the Jacobi identity in  .

Explicitly, the differential of the  -cochain   is the  -cochain   given by:[3]

 

where the caret signifies omitting that argument.

When   is a real Lie group with Lie algebra  , the Chevalley–Eilenberg complex may also be canonically identified with the space of left-invariant forms with values in  , denoted by  . The Chevalley–Eilenberg differential may then be thought of as a restriction of the covariant derivative on the trivial fiber bundle  , equipped with the equivariant connection   associated with the left action   of   on  . In the particular case where   is equipped with the trivial action of  , the Chevalley–Eilenberg differential coincides with the restriction of the de Rham differential on   to the subspace of left-invariant differential forms.

Cohomology in small dimensionsEdit

The zeroth cohomology group is (by definition) the invariants of the Lie algebra acting on the module:

 

The first cohomology group is the space Der of derivations modulo the space Ider of inner derivations

 ,

where a derivation is a map   from the Lie algebra to   such that

 

and is called inner if it is given by

 

for some   in  .

The second cohomology group

 

is the space of equivalence classes of Lie algebra extensions

 

of the Lie algebra by the module  .

Similarly, any element of the cohomology group   gives an equivalence class of ways to extend the Lie algebra   to a "Lie  -algebra" with   in grade zero and   in grade  .[4] A Lie  -algebra is a homotopy Lie algebra with nonzero terms only in degrees 0 through  .

See alsoEdit

ReferencesEdit

  1. ^ Cartan, Élie (1929). "Sur les invariants intégraux de certains espaces homogènes clos". Annales de la Société Polonaise de Mathématique. 8: 181–225.
  2. ^ Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie". Bulletin de la Société Mathématique de France. 78: 65–127. Archived from the original on 2019-04-21. Retrieved 2019-05-03.
  3. ^ Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge University Press. p. 240.
  4. ^ Baez, John C.; Crans, Alissa S. (2004). "Higher-dimensional algebra VI: Lie 2-algebras". Theory and Applications of Categories. 12: 492–528. Archived from the original on 2018-07-20. Retrieved 2019-05-03.

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